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Theorem elcnfn 28869
 Description: Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elcnfn (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝑇

Proof of Theorem elcnfn
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6228 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑤) = (𝑇𝑤))
2 fveq1 6228 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
31, 2oveq12d 6708 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡𝑤) − (𝑡𝑥)) = ((𝑇𝑤) − (𝑇𝑥)))
43fveq2d 6233 . . . . . . 7 (𝑡 = 𝑇 → (abs‘((𝑡𝑤) − (𝑡𝑥))) = (abs‘((𝑇𝑤) − (𝑇𝑥))))
54breq1d 4695 . . . . . 6 (𝑡 = 𝑇 → ((abs‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦 ↔ (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦))
65imbi2d 329 . . . . 5 (𝑡 = 𝑇 → (((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
76rexralbidv 3087 . . . 4 (𝑡 = 𝑇 → (∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
872ralbidv 3018 . . 3 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
9 df-cnfn 28834 . . 3 ContFn = {𝑡 ∈ (ℂ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}
108, 9elrab2 3399 . 2 (𝑇 ∈ ContFn ↔ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
11 cnex 10055 . . . 4 ℂ ∈ V
12 ax-hilex 27984 . . . 4 ℋ ∈ V
1311, 12elmap 7928 . . 3 (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
1413anbi1i 731 . 2 ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)) ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
1510, 14bitri 264 1 (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   class class class wbr 4685  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  ℂcc 9972   < clt 10112   − cmin 10304  ℝ+crp 11870  abscabs 14018   ℋchil 27904  normℎcno 27908   −ℎ cmv 27910  ContFnccnfn 27938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-hilex 27984 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-cnfn 28834 This theorem is referenced by:  cnfnc  28917  0cnfn  28967  lnfnconi  29042
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